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What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$ with an error term $E(T) = O(T^{\alpha+\epsilon})$ with $\alpha=1/2$ or $\alpha=1/3$?

Note I am not asking for the best $\alpha$; I am asking for different ways - the simpler the better - with the intention of choosing one of them to make explicit.

Methods I know:

  • Proofs based on Atkinson's 1949 formula. This formula yields $\alpha=1/2$ automatically and $\alpha=1/3$ after some smoothing and some work. The problem is that there is an asymptotically tiny error term that seems hard to make explicit without applying the saddle-point method explicitly (twice) - something that I know to be time-consuming and sometimes painful.

  • Balasubramanian (1978). Again, this doesn't look simple.

  • Ingham/Atkinson (1939). Short and relatively simple. Yields $E(T) = O(T^{1/2} \log^2 T)$.

I'm also curious about the same question (in the same way) for $$\int_0^T \left|\zeta\left(\sigma + i t\right)\right|^2 dt,$$ where $0\leq \sigma <1$.

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Theorem 7.2(A) of Titchmarsh's 'The Theory of the Riemann zeta function' uses a simplified form of the special case $x=t$ of the approximate functional equation of the Riemann zeta function which holds for $1/2<\sigma\leq 1$, $$ \zeta(s)=\sum_{n<t} n^{-s} + O(t^{-\sigma}). $$ The result is $$ \int_1^T|\zeta(\sigma+it)|^2 \ dt< AT \min\left(\log T, \frac1{\sigma-\frac12}\right) $$ uniformly for $1/2\leq \sigma \leq 2$.

For $0<\sigma<1/2$, the above result and the functional equation may help.

More precisely, for $\sigma>1/2$, Theorem 7.2 of the same book gives $$ \int_1^T |\zeta(\sigma+it)|^2 \ dt \sim \zeta(2\sigma) T. $$

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  • $\begingroup$ That's not what I would call an approximate functional equation, and I was looking for more than just an upper bound. $\endgroup$ – H A Helfgott Jan 15 '20 at 1:16
  • $\begingroup$ If that's the case, Theorem 7.2 gives an asymptotic formula, not 7.2(A). $\endgroup$ – Sungjin Kim Jan 15 '20 at 1:18
  • $\begingroup$ Again, see my post for the rough strength I am aiming for. Incidentally, Theorem 7.2 is valid only for sigma>1/2. $\endgroup$ – H A Helfgott Jan 15 '20 at 1:21

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